15 WEAK TOPOLOGIES 75 v is a continuous linear mapping of F into a locally convex space G, then (12.15.6) '(z; o w) = 'w o '*;. The weak topology on the dual of a locally convex space is not metrizable in general (Problem 2). However, there is the following result: (12.15.7) Let E be a separable metrizable locally convex space. IfH is any equicontinuous (7.5) subset of the dual E' o/E, then the weak closure ofH in E7 is a compact metrizable space in the weak topology. We shall first prove the following lemma: (1 2.1 5.7.1 ) Let E be a metrizable vector space and F a normed space. In order that a set H of linear mappings ofE into F should be equicontinuous, it is neces- sary and sufficient that there should exist a neighborhood V of® in E and a real number c> 0 such that \\u(x)\\ £ cfor all x e V and all ueH. (If E is normed, this condition is also equivalent to sup \\u\\ < H-oo weH by (5.7.1).) The condition expresses that H is equicontinuous at the point 0. If this is the case, then for each x0 e E and each x e x0 4- eV, we have \\U(X)-U(XO)\\ = \\U(X-XO)\\£BC for all u e H, and therefore H is equicontinuous at x0 . In proving (12.15,7), we may first of all restrict ourselves to the case where H is weakly closed in E'. For if fl is the weak closure of H in E', the continuity of the mapping X'H><JC, x") on E' implies that if |<X *')| ^ c for all xe Vand all xf e H, then also |<x, x')\ £ c for all jc e V and all x' e H (3.15.4), whence the result by (12.15.7.1). Now let (an) be a sequence which is dense in E. We shall show that the weak topology on H is defined by the pseudo-distances Clearly the topology defined by these pseudo-distances is coarser than the weak topology; hence it is enough to show that it is also finer. By hypothesis, every neighborhood of a point x'0 e H in the weak topology contains a subset W of H consisting of elements x' e H satisfying & finite number of inequalities (12.15.7.2) |<jt,, x' - *i>| < r (l£i£m)k topology.